LISA Double Black Holes: Dynamics in Gaseous Nuclear Discs

arXiv:astro-ph/0509813v2 12 Dec 2005

o.Nt .Ato.Soc. Astron. R. Not. Mon. 1Spebr2018 September 11 2 1 asm Dotti Massimo LISA vlto fglxe n asv lc oe MH)ex- (MBHs) from ranging holes masses with black (BHs), massive holes Black and and ists. formation galaxies the between of link evolution tight a that know we Today, INTRODUCTION 1 o cetfi agt ftenx ae nefrmtrSpac ma- Interferometer the Laser next of the one of is targets scientific emission jor whose radiation, gravitational history. cosmic the during have number must large (MBHBs) in binaries forma formed hole im- structure been black as of massive merge models then galaxies MBHs tion, clustering if If and hierarchical past, 2000, 2004). by the al. plied Rix in et stellar Häring common & also Gebhardt host 2000, were 1998, the Merritt al. of & et Ferrarese properties Magorrian the show (e.g., masses with their bulges correlation and galaxies, tight nearby a of centre the in tous o10 to c iatmnod iia”.Ociln” nvri´ iMil Universitá di Occhialini”, ”G. Fisica Universitá dell’Ins di Matematica, Dipartimento & Fisica di Dipartimento 00RAS 0000 ls BB r aua,vr oeflsucsof sources powerful very natural, are MBHBs Close 9 M ⊙ obebakhls yaisi aeu ula discs nuclear gaseous in Dynamics holes: black double eg,Kred ihtn 95,aeubiqui- are 1995), Richstone & Kormendy (e.g., 1 oiaColpi Monica , 000 0–0 00)Pitd1 etme 08(NL (MN 2018 September 11 Printed (0000) 000–000 , h rvttoa oqeeetdo h iayb urudn ga surrounding by ( probed binary scale the smallest the on to exerted down momentobservable torque angular eccentricity, gravitational initial their the and subsid regardless decrease, has and friction not holes dynamical black When does binary. case eccentric present) an the (if forming In grav eccentricity bind coalescence. hinder initial to binary the may the ga orbits res leading the Circularization and in numerical over for operates. present taking our bind rotation from friction the within holes dynamical in zero black resides where with circularization massive of consistent the they cause eccentricity, if consequence The low eccentricity a orbital a As initial with their disc. t of in gaseous memory that the find dyn lose holes We binary. black holes hole close the black a black on form the they acts the until friction simulation follow our gas–dynamical continue we when simulations, phase early SPH the high–resolution s Using rotationally circum–nuclear, disc. massive a inside orbiting detectability, ABSTRACT obeatv ulimyfr hncruaiaini opee,o dis on words: completed, Key is circularization parsecs. when of pair form tens hole weak black may of and the nuclei during occu velocities active excited relative holes be double may black high activity har AGN the imply further Thus, by orbits focusing. may gas eccentric that of orbits: distribution capture circular th gas gravitational and ellipsoidal limit, inspiral, any resolution During our of to formation down of efficient remains friction dynamical uli–GaiainlWvs–Qaa:general Quasar: – Waves Gravitational – nuclei esuyteisia fdul lc oe,wt assi the in masses with holes, black double of inspiral the study We 2 rnec Haardt Francesco & , ∼ lc oePyis iais yrdnmc aais evolution, Galaxies: – hydrodynamics binaries, Physics: Hole Black 10 6 bi,VaVlego1,210Cm,Italy. Como, 22100 11, Valleggio Via ubria, n–ioc,Paz el cez ,210Mln,Italy. Milano, 20100 3, Scienze delle Piazza ano–Bicocca, M ⊙ e - ntelwredo h bevdms itiuin n are and distribution, study. mass our observed of the target of the end lower the in aswno ensor“ our defines window mass e ta.19) estv ntefeunyrnebetween range frequency the in sensitive 1994), 10 al. et der bet (10 objects h asrne10 range mass the nen ( Antenna ilr&Clet20) ihrmass Higher 2005, 2004). Ho & Colbert Gua- Rich & Gebhardt, & 2002, Miller Possenti al. Colpi, et Gerssen 2002, 2002, a Ho landris for & 2002 Rich al. Gebhardt, et recently, Marel review, and, forma- der 2003), (van structure support Madau observational of & gained Haardt models Volonteri, in (e.g., hierarchical conjectured tion been of has framework existence the their BHs: mass mediate 03 eaae l 05.The 2005). al. et Sesana 2003, − 5 LISA − 10 − ildtc BB nydrn h ato com- a of last the during only MBHBs detect will LISA 3 1 − z iluvi,we noeain BB in MBHBs operation, in when unveil, will Hz, 5 M 1 e,eg,Henl 94 ae&Backer & Jaffe 1994, Haehnelt e.g., see, ; ⊙ ≃ 3 A / T M 1+ (1 c.I h aeo nqa masses, unequal of case the In pc). 1 E ⊙ tl l v1.4) file style X / 1+ (1 z )aeotnrfre oa inter- as to referred often are )) LISA z LISA ) ∼ < M lc oe” h lighter The holes”. black BH nefrmtr(e Ben- (see interferometer LISA ∼ < h lc oe may holes black the fcounter–rotating of 10 mls,die by driven loss, um eu background seous d o qa mass equal for ed, potdgaseous upported ,i nevertheless is s, niiuly and individually, 7 LISA smil along mainly rs e h binary. the den eerysinking early he M ttoa waves itational n rcs and process ing H r instead are BHs igabinary a ming ortt with co–rotate lto limit. olution ⊙ r sn sign no is ere gravitational tance–scales / 1+ (1 idwof window mc in amics z :this ): 2 Dotti, Colpi & Haardt plex sequence of events that starts when the two MBHs are BH pairing. Recently, Kazantzidis et al. (2005) explored a few kpc far apart, and terminates when they reach sub–pc the effect of gaseous dissipation in mergers between gas– scales, i.e., the distance at which gravitational waves (GWs) rich disc galaxies with central BHs, using high resolution ultimately drive the final coalescence. How can MBHs reach N–Body/SPH simulations. They found that the presence of the GW emission regime? The overall scenario was first out- a cool gaseous component is essential in order to bring the lined by Begelman, Blandford & Rees (1980) in their study BH to close distances, since gas infall deepens the potential of the long–term evolution of BH pairs in dense stellar sys- well, preserving the less massive galaxy against tidal dis- tems. They indicated three main processes for the loss of ruption. Moreover, the interplay between strong gas inflows orbital energy and angular momentum: (a) dynamical fric- and star formation seems to lead naturally, in these mergers tion against the stellar background acts initially on the BHs and regardless of the masses of the interacting galaxies, to as individual masses, favoring their pairing; (b) MBHs even- the formation, around the two MBHs, of massive circum– tually bind to form a binary when the stellar mass enclosed nuclear gaseous discs on a scale ∼ 100 pc, close to the nu- in the orbit becomes less than the total mass of the two merical resolution limit. Yet, it is still difficult to establish MBHs. The resulting binary continues to harden via 3–body the internal kinematic properties of the disc and of the MBH interactions with the surrounding stars until it reaches the orbits, whether they are elongated or circular. separation at which GWs become dominant; (c) in the third, The work of Kazantzidis et al. (2005) and the obser- and last, phase, GW back—reaction shrinks the binary and, vational evidence that discs are ubiquitous, have provided depending on the eccentricity and separation of the orbit, our main motivation to study the process of BH pairing in leads to coalescence. The dynamical range that a MBHB gaseous circum–nuclear discs. Escala et al. (2005, hereinafter separation needs to cover to become a LISA source is enor- ELCM05; see also Escala et al. 2004) have studied the role mous, more than six orders of magnitude. played by gas in affecting the dynamics of MBHs of equal 7 9 Early studies have explored the pairing of MBHs in masses (in a range of 5 × 10 M⊙ ≤ MBH ≤ 2.5 × 10 M⊙) mergers of purely collisionless spherical halos (Makino & moving on circular orbits in Mestel discs of varying clumpi- Ebisuzaki 1996, Milosavljević& Merritt 2001, Makino & ness. They explored the decay across phase (a) controlled Funato 2004). Governato, Colpi & Maraschi (1994) first no- by dynamical friction and the transition to the regime (c) ticed that when two equal mass halos merge, the MBHs dominated by GW emission, highlighting the role played by inside their host nuclei are dragged toward the centre of the gravitational torques in shrinking the binary. ELCM05 fol- remnant galaxy, forming a close pair. The situation is dif- lowed the orbital decay of a MBHB of total mass equal to 8 ferent in unequal mass mergers, where the less massive halo 10 M⊙ down to 0.1 pc, just around the critical distance for is tidally disrupted, and leaves its MBH wandering in the the transition of the GW domain. The last phase of orbital outskirts of the main halo. Thus, depending on the mass ra- decay in a gaseous enviroment has been studied by Armitage tio and internal structure of the halos and host galaxies, the & Natarajan (2002), and by Milosavljevic & Phinney (2005). transition from phase (a) to phase (b) can be prematurely The first paper addresses the issue of black hole migration aborted. Similarly, the transit from phase (b) to phase (c) is within a pre–exisiting Shakura & Sunayev (1973) accretion not always secured, as the stellar content of the “loss cone” disc around a much heavier black hole. It is show how orbital may be not enough to drive the binary separation to the GW angular momentum losses due to binary–gas interactions can emission regime (see, e.g., Milosavljevic & Merritt 2001, Yu shrink the orbit in ∼ 107 yrs, down to a separation where 2002, Berczik, Merritt & Spurzem 2005, Sesana, Haardt & GW emission rapidly leads the binary to coalescence. The Madau 2005, in preparation). In the studies cited above, the authors suggest that an increased accretion rate during the background was purely collisionless. migration is unlikely, while strong, potentially observable Since LISA BH coalescences are likely to be events asso- outflows should preceed the very last phase of the evolution. ciated with mergers of galactic structures at high redshifts, Milosavljevic & Phinney (2005) argued that, after coales- it is likely that their dynamics occurred in gas dominated cence, residual disc gas could fill the circum–binary gap, backgrounds. So one might expect that phases (a) and (b) producing, within few years, an X–ray LISA afterglow. can be profoundly affected by the presence of a dissipative Further studies are necessary. First, because LISA is component. Mergers cause large–scale gas dynamical insta- designed to detect MBHBs lighter than the mass range ex- bilities that lead to the gathering of cool gas deep in the plored by ELCM05, and, second, because, in hierarchical potential well of the interacting systems dragging the BHs cosmologies, only mergers occurring at very high redshifts to the centre of the remnants. involve almost equal mass MBHBs, while, at later epochs, Observations of interacting Luminous Infrared Galax- coalescences of unequal mass MBHBs are much more com- ies (LIRG) in our local universe have provided evidence of mon (Volonteri et al. 2003). In the present work, we in- the presence of huge amounts of cool atomic and molecular vestigate the sinking process of LISA BHs inside a massive 8 hydrogen collected in their cores (Sanders & Mirabel 1996). rotationally supported disc (MDisc ∼ 10 M⊙) having finite 9 The total gas mass is typically ∼ 5 × 10 M⊙, located in the vertical extension. The disc is constructed starting from an rotationally supported disc in the inner ∼ 100 pc (Downes initial Mestel distribution, and is allowed to relax into a & Solomon 1998). Interestingly, at least in three cases (NGC new dynamical equilibrium. The BHs have equal and un- 6240, Arp 299 and Mkn 463) combined X–ray and infrared equal masses and are moving initially on orbits of varying observation hint for the presence of two active MBHs in their eccentricity: from circular to highly elongated, in order to nuclei (Hutchings & Neff 1988, Komossa et al. 2003, Ballo reflect different conditions in their sinking from the kpc– et al. 2004). distance scale down to the scale of the nuclear disc which On theoretical ground, the advances in numerical com- they inhabit. puting allow to investigate in greater detail the process of We would like to address a number of questions.

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8 5 (i) How do eccentric orbits evolve? Do they become cir- units of the code are: [Mass]=10 M⊙, [Time]=2.5 × 10 yr, cular? This issue may be relevant in establishing the initial [Length]=10.9 pc, [Velocity]=41.5kms−1, [Density]=7.8 × 4 −3 conditions for the braking of the binary (phase b) due to the 10 M⊙ pc . In these units Vcir = 3.7 (in the limit of zero slingshot mechanism and/or gaseous gravitational torque. disc thickness), and G = 21.9606. Armitage & Natarajan (2005) have recently shown that a The equation of state for the gas in the disc is a poly- residual small (but non zero) eccentricity can be amplified trope, by the binary–disc interaction before GW emission domi- γ nates. P = Kρ , (3) (ii) During the sinking process, do the BHs collect sub- where γ is equal to 5/3, corresponding to pure adiabatic stantial amounts of gas? This is a query related to the poten- evolution, and K is set equal to 2.3325, the value considered tial activity of a MBH during a merger and its detectability by ELCM05, for which the level of clumpiness during the across the entire dynamical evolution. evolution is minimized. In internal units a = 5, MBulge = (iii) When a binary forms, how gravitational torques, 6.98, RDisc = 10, MDisc = 1 and the sound speed of gas in exerted by the surrounding gas, depend on the mass ratio 0.5 the disc at r = 5 is cs =(∂P/∂ρ) = 0.289, corresponding between the two BHs? The possibility that the binary stalls to a temperature of T ≃ 104K. In our scheme we have not has to be studied with more scrutiny and may also depend included the possibility that the gas develops a multiphase on the issue (i). structure. This has been recognized as a relevant feature of (iv) What mechanisms, besides this torque, can drive self–gravitating discs (see, e.g., Wada & Norman 2001), and the binary into the GW dominated decaying phase? We con- it is related to various feed–back mechanisms operating in sider the effects of BH mass growth on their dynamics in §5. realistic situations (e.g., star formation, radiative heating, The paper is organized as follows. In section §2 we de- shock induced cooling, etc.). Though a simple polytropic scribe the numerical simulations we performed. Results con- model, as the one we employed, does not catch the detailed cerning equal mass binaries are reported in §3, while in next thermodynamics of the gas, it is, nevertheless, an effective §4 we describe unequal mass systems. Last §5 is devoted to tool to study the orbital decay of MBHBs in self–gravitating discussion and conclusions. discs (ELCM05). In our simulations, the number of collisionless particles is 105, for the spheroid, while the number of gas/SPH 2 N–BODY/SPH SIMULATIONS particles is 235,331. With the above figures, our gas mass resolution is 100 times the mass of a single SPH particle, The circum–nuclear region of a merger remnant is described −6 which is, in internal units, 4.25 × 10 (425 M⊙ in physical here as a superposition of a spherical stellar spheroid and −5 units). The mass of bulge stars is 6.98×10 (6980 M⊙), and of an axisymmetric gaseous disc. The disc hosts two BHs the softening length is 0.1 (1.09 pc), equal for both type of treated as collisionless particles. The spheroidal component particles. (bulge) is modeled initially as a Plummer sphere, while the Given these initial conditions, the collisionless spherical disc as a Mestel distribution. We evolve the system using the component, drawn from the Plummer phase–space distribu- N–Body/SPH code GADGET (Springel, Yoshida & White tion function, is in near equilibrium, while the disc, having 2001). finite thickness and homogeneous vertical density distribution is allowed to evolve into an equilibrium configuration 2.1 Bulge and gaseous disc along the R and z–axis. The disc is evolved for a dimensionless time of ∼ 10 (2.6 Myrs) until the density and pressure The Plummer law for the stellar bulge density is fields find their equilibrium. Initially, the vertical collapse −5/2 of the gas increases the pressure gradient in both vertical 3 M r2 Bulge and horizontal directions, exciting small waves that prop- ρ(r)= 3 1+ 2 , (1) 4π a a agate outwards. In settling toward equilibrium, Σ modifies where a is the core radius, r the radial coordinate, and and a small core forms in the central region, as shown in MBulge the total mass of the spheroid. Figure 1 where we draw the initial (dashed line) and equi- The Mestel disc follows a surface density profile librium (solid line) surface density profile; the gas density in the z direction becomes non uniform, when equilibrium is Σ0 R0 ΣDisc(R)= (2) attained. R where R is the radial distance, projected in the disc plane, and Σ0 and R0 reference values. The disc is rotationally 2.2 Black holes supported and is characterized by a circular velocity Vcir independent of R, in the limit of infinitesimal thickness The BHs are treated as collisionless particles and are placed and low temperature. The bulge, introduced to stabilize the in the disc plane. This is an assumption in agreement with disc against gravitational instabilities, has a mass MBulge = the large scale simulation by Kazantzidis et al. (2005) who 6.98MDisc. find that the BHs pair inside the massive circum–nuclear In our current hydrodynamical simulations, the disc disc which forms in the remnant galaxy, on elongated orbits. has finite radial extension RDisc = 2a, and finite vertical We place our BHs on circular as well as rather eccentric thickness Z0 equal to a tenth of the disc radius RDisc. The orbits to bracket uncertainties. 6 vertical density profile is uniform, for a given R, initially. We consider the case of LISA BHs with 1:1 (10 M⊙ − 6 6 6 The total disc to bulge mass ratio inside RDisc is 1:5. The 10 M⊙) and 5:1 (5 × 10 M⊙ − 10 M⊙) mass ratio. The

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Figure 1. Surface density Σ as a function of the distance R from Figure 2. Face–on projection of the disc for run A at time 2 the centre of mass, in internal dimensionless units. The dashed Myrs. The color coding shows the z–averaged gas density, and line refers to the surface density of the Mestel disc, while the the bright dots highlight the position of the two BHs that form solid line describes the equilibrium proﬁle after an elapsed time prominent wakes behind their trails. ≃ 10 (2.6 Myrs), corresponding to the initial condition of all our simulations.

We plot in Figure 2 the density map of the gas surrounding the BHs at a selected time, to show the prominent Table 1. Run parameters over-densities that develop behind the holes causing their braking. The motion of the BHs is highly supersonic, and this explains the coherent structure and shape of their wakes ∗ ∗ ∗ ∗ run MBH1 MBH2 MDisc MBulge e (Ostriker 1999). Most of the disc gas lying out of BH orbits is somewhat “squeezed” into the wakes. Qualitatively, the extent of the wakes depends on the amount of disc mass perturbed by the orbiting BHs, which is a function of the A 1 1 100 698 0 BH masses. B 1 1 100 698 0.97 C 5 1 100 698 0 Figure 3 shows the inspiral of the two BHs sinking be- D 5 1 100 698 0.95 cause of dynamical friction mainly due to the gas compo- E 1 1 0 698 0.94 nent. The BHs evolve maintaining their orbits nearly cir- F∗∗ 5 1 100 698 0.95 cular until they reach the central, numerically unresolved, region. The time evolution of the BH relative separation Rrel is plotted in the inset of Fig. 3, and is in agreement ∗ 6 Masses are in units of 10 M⊙. with ELCM05. ∗∗ BH2 in run F has a retrograde orbit. Before reaching the force resolution limit, where integration stops, orbital decay becomes less efficient since the softening of the collisionless BH particles is 0.1 (1.09 pc). In nature of the drag changes. Figure 4 gives, at late times, the Table 1 we list the parameters used in our 6 simulations. mass in star and gas enclosed inside the sphere defined by > Rrel. When such mass is ∼ 2 MBH, dynamical friction acts on the BHs as individual objects. At a time t ≃ 8 Myrs the mass drops below 2 MBH and dynamical friction becomes 3 DYNAMICS OF EQUAL MASS BLACK inefficient. The BHs now are bound in a “binary”, and the HOLES orbital decay proceeds further, but at a lower pace. This corresponds to the “transition regime” defined by ELCM05. 3.1 Circular Orbits Our resolution limit does not permit to test the subsequent Run A is aimed to reproduce run D of ELCM05. Each LISA “ellipsoidal regime” (ELCM05). We note that an ellipsoidal BH has a mass MBH = 0.01MDisc and is set on a circular distribution of gas is already present when the binary is prograde orbit inside the disc. The initial separation, relative formed, as shown in Figure 8. to the centre of mass, is 5 (54.5 pc). The BHs maintain a nearly circular orbit and this im-

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plies that the relative velocity between the BHs and the rotationally supported gas particles is small. This suggests that some of the gas in the wake may become bound to the BHs while spiraling inwards. We ﬁnd that the mass in gas particles associated to the over-density centred around each BH is ∼ 20% of its mass, during inspiral. This number is cal- culated by summing the masses of all gas particles associated with a spherical density excess measured relative to the un- perturbed disc, at the same position. This provides only an approximate estimate of the mass that can become bound to the BHs. The poor resolution close and inside the BH sphere of inﬂuence, and the simple thermodynamics used, prevent us from giving an estimate of the mass accretion rate on the horizon distance-scale. We can only speculate that in this phase the BH may accrete, generating a double nucleus AGN.

3.2 Eccentric orbits We consider here the case of two equal mass BHs in the disc plane, the first moving on a initially circular orbit, and the second on an initially eccentric orbit (e = 0.97) with same binding energy (run B). Figure 5, upper panel, shows Figure 3. Distance R from the centre of mass, as a function of the BH distances from the centre of mass, as a function of time t, for equal mass BHs in run A. Solid and dashed lines refer time. We find that the sinking time of the eccentric BH is to the different BHs. The insert gives the BH relative separation comparable to that of the companion, but what is remark- R versus time. Integration is halted when the resolution limit rel able is the strong effect of circularization seen in the orbit of (≃ 1 pc) is attained. the eccentric BH. Dynamical friction in a rotationally supported gaseous medium makes eccentric sinking orbits circular. This is opposite to what is found in isotropic, purely collisionless spherical backgrounds (Colpi, Mayer & Gover- nato 1999, van den Bosch et al. 1999), or in a spherical pressure–supported gaseous background (Sanchez–Salcedo & Brandenburg 2000). Figure 6 shows the density map of the gaseous disc viewed face–on. During the sinking process the BHs develop prominent wakes that are perturbing the underlying gas density. The BH moving initially on a circular orbit, spirals inwards maintaining its over-density behind its trail. The BH moving initially on the eccentric orbit undergoes instead a remarkably different evolution. Close to the pericentre (upper left panel) the eccentric BH has a speed larger, in mod- ulus, than the local gas speed, and so its wake is excited behind its trail. The wake brakes the orbit and erodes the radial component of the velocity. On the other hand, around the apocentre the BH moves more slowly and its tangential velocity (which dominates over the radial, at this distance) is lower that the local rotational gas velocity. This causes the interesting fact, clearly illustrated in the upper right panel, that the wake is dragged in front of the BH, increasing its angular momentum. When approaching again pericentre, the wake tends to realign behind the BH, as shown in the two lower panels. The net effect highlighted in Figure 6 is the circularization of the BH orbit (see Figure 5, upper panel). Figure 4. Upper panel: relative distance Rrel between the BHs versus time, for run A. Middle (lower) panel: mass (in internal We have run a case without disc, letting the BHs sink units) in stars (gas) enclosed inside Rrel, against time. Horizontal under the action of the drag force due solely to the stellar dashed lines indicate the total mass of the BHs. bulge (run E). As illustrated in Figure 7, we find that the inspiral takes a longer time compared to the case with gas, since the density of stars does not rise significantly (in a Plummer model), but, interestingly, the drag force does not lead to any circularization of the orbit, due to the lack of rotation in the

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Figure 6. Time sequence of the sinking of the BHs, from run B. The panels show a face–on projection of the disc and BH positions at four diﬀerent times. The color coding indicates the z–averaged gas density (in linear scale), and the white lines trace the BH counterclockwise prograde orbits. In the upper left panel the over–density created by both BHs are behind their current trail, while in the right upper panel, the BH moving on the eccentric orbit ﬁnds its own wake in front of its path. The wake is dragged by the faster rotation of the disc. In the two lower panels we observe a bending of the wake that tends to re–align, with time, behind the direction of motion of the BH.

background. Furthermore, we observe an increase in the ec- substantial gas mass in its vicinity, given its high velocity centricity in the BH moving initially on a circular orbit, in relative to the underlying background. Only when the orbit line with the ﬁndings of Colpi et al. (1999). becomes circular the gathering of gas can occur (see Fig- ure 5, lower panel). Thus, double nuclear activity does not As far as accretion is concerned, we notice that the BH moving along the initially eccentric orbit is unable to collect

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Figure 5. Upper panel: Solid (dashed) line shows the distance R Figure 7. Distance R from the centre of mass, as a function of (pc) of the eccentric (circular) BH from the centre of mass of the time t, for run E. In the simulation the sole collisionless bulge is system as a function of time t. Lower panel: Solid (dashed) line considered. Solid (dashed) line refers to the BH set initially onto shows the mass of the over-density in internal units (as deﬁned an prograde eccentric (circular) orbit. In the insert we give the in the text) corresponding to the eccentric (circular) BH as a relative orbital distance; axes are in the same units. function of time.

always set in, but instead it depends on the properties of the BH orbits. When circularization is completed, the sinking process ends as in run A. Figure 8 shows the density map, in the plane of the BHs zBH, at the time the two BHs form a binary system, i.e., when the action of dynamical friction becomes ineﬃcient; an over-density of ellipsoidal shape surrounds the binary, resulting from the superposition of the gravitational potentials of the two BHs.

4 DYNAMICS OF UNEQUAL MASS BLACK HOLES Since merging galaxies may host BHs with diﬀerent masses, in this Section, we explore the dynamics of two BHs with mass ratio 5:1 (see Table 1 for the details of the runs). The heavier BH has a mass MBH1 = 0.05MDisc.

4.1 Circular orbits We first explored the case in which the two BHs move initially on circular prograde orbits, at equal distances from Figure 8. Density profile in z = zBHs plane at time 15 Myrs. the centre of mass (run C). The large, more massive BH Lines describe isodensity regions of ρ = 0.25, 0.125, 0.09375 and sinks rapidly toward the centre of the gaseous disc, over a 0.0625 (in internal units). Black dots correspond to the two BH timescale of ∼ 4 Myrs, as illustrated in Figure 9, while the positions. lighter completes its orbital decay on a timescale longer by a factor ≃ 2 − 3. This implies that the dynamical friction time does not scale exactly as the inverse of the mass, and we interpret this result as an effect related to the pertur- bation in the overall disc gravitational potential caused by

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Figure 9. Same as Figure 3 for run C. Solid (dashed) line refers Figure 10. Same as Figure 6 for run C. In the upper panel to the lighter (heavier) BH. (t = 1 Myr) the heavier BH perturbs the gaseous disc and shifts the barycentre of the system away from the lighter one. In the lower panel (t = 3.5 Myr), the heavier BH is located between the other BH and the barycentre, speeding up the orbital decay of the lighter one.

the larger BH. During the early stages, the two BHs are symmetrically displaced relative to the centre of mass, and 4.2 Eccentric orbits start to develop their own wakes of different intensity. In In run D, the light BH is initially set on a prograde orbit the upper panel of Figure 10 we catch the instant at which with e = 0.95. Its sinking as a function of time proceeds, sim- the heavier BH, having perturbed the gaseous mass, causes ilarly to run B, with the circularization of the orbit. Since a shift of the barycentre, i.e., a displacement in the direc- the larger BH is already in place at the centre of the gaseous tion opposite to the lighter BH. At this time, the orbital disc we do not see any effect on the acceleration of the or- decay of the less massive BH halts temporarily (as shown bit besides dynamical friction. As shown in the upper panel also in Figure 9 between 3 and 6 Myrs). When the orbital of Figure 11 the orbit of the light hole becomes circular decay of the heavy BH is sufficiently advanced that it has at a time t ≃ 5 Myrs. The evolution of the eccentricity is almost reached the centre, its location is in between the po- shown in Figure 12. In the early inspiral, the gas mass asso- sition of the lighter BH and the barycentre which is now ciated to the over-density in the neighborhood of the small −3 shifted toward the light hole deepening the potential well. BH is only MGas ∼ 3 × 10 MBHl , and only when the or- This causes an acceleration toward the disc centre speeding bit becomes circular, we observe a rapid increase to a value the orbital decay (see lower panel of Figure 10 and Figure 9 ∼ 0.25MBHl , potentially triggering an episode of accretion. at times ∼> 6 Myrs). This case illustrates that time varia- When the light hole binds to the larger one (t ≃ 11 Myrs), tions in the underlying gravitational potential, that can be it finds itself embedded inside the over-density created by computed self–consistently in a real simulation, can modify the large hole, as shown in the lower panel of Figure 11. granted dependences of the dynamical friction timescales. Contrary to the case of equal mass BHs, at very late times, We find also that dynamical friction is effective until we hit the overdensity distribution around the binary BHs is not the force resolution limit, and that no ellipsoidal density any longer ellipsoidal in shape and the gravitational field is distribution is found (see §4.2 for a discussion). weakly dipolar. Figure 13 shows how strong is the degree of Our initial condition is clearly arbitrary, and it is likely sphericity of the gas surrounding the two BHs. Does orbital that in real mergers the heavier BH is already in place at decay proceed further? The light BH seems to decay but the centre of the circum-nuclear disc by the time the second at a much lower pace. The lack of a visible wake and the BH enters the disc. For this reason the sinking process may absence of an ellipsoidal deformation (torquing the binary be different for the same BH masses involved, depending on components) suggest that the gas has become sufficiently the details of the process of paring. In the next simulation stiff and the potential well sufficiently deep, due to the pres- where the lighter BH is set onto an eccentric orbit, we allow ence of the more massive hole, that orbital decay is halted the more massive hole to reach the centre before the sinking or delayed. process of the light one takes place. In run F, we explore the dynamics of an unequal mass

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Figure 11. Upper panel: Solid (dashed) line shows the distance Figure 12. Eccentricity of the less massive BH as a function of R (pc) of the lighter (heavier) BH from the centre of mass of the time, in the case of prograde (Run D, triangles), and retrograde system as a function of time t. Lower panel: Solid (dashed) line (Run F, squares) orbit. The eccentricity is computed over any shows the mass (in internal units)of the over-density correspond- half orbit. ing to the lighter (heavier) BH as a function of time.

BH set on a retrograde eccentric orbit. This is opposite to run D and is considered in order to bracket uncertainties in the way BHs bind. We ﬁnd that the orbit remains eccentric (see Figure 12), preventing the gas to accumulate substantially around the BHs during the whole inspiral process. Given the rotational pattern of the gas, the over-density created by the BH stays always behind its trail, so that the eccentric BH does not circularize, as shown in ﬁgure 14. Note that the sinking time is about twice larger if compared to the prograde cases.

5 DISCUSSION We explored the dynamics of a MBH pair orbiting inside a gaseous disc embedded in a spherical stellar distribution. We followed the slow inspiral of the pair, driven by dynamical friction, until the MBHs bind to form a close binary. The calculation is idealized in many ways, since we assumed a particular density distribution for the gas particles (Mestel disc), and neglected gas cooling and star formation. Despite these limitations, we highlighted basic features of the dynamics of LISA double BHs in gas–rich discs. When Figure 13. Same as Figure 8 for run D at time 13 Myr. The a BH initially is moving on a highly eccentric co–rotating or- lines describe regions of ρ = 2, 1, 0.5, 0.25, 0.125, 0.0625 (in inter- bit, its eccentricity decreases signiﬁcantly, contrary to what nal units). The isodensity contours are almost circular. occurs when the background is spherical and collisionless (as shown in Figure 7; Colpi et al. 1999). Disc rotation is the key element of the circularization: near apocentre, where the angular velocity of the MBH is smaller than that of the disc, the density pattern, created by the MBH along its motion, is dragged in front of the BH itself enhancing its angular momentum. As a consequence the MBHs tend to form a close

c 0000 RAS, MNRAS 000, 000–000 10 Dotti, Colpi & Haardt

two masses, the gas distribution around the binary remains remarkably spherical and there is no gravitational torque in action to cause further inspiral of the light BH, within our resolution limit. The feeding of BHs during their inspiral is also a key related issue. Despite the limitations of the thermodynamics employed, we have shown that a MBH moving on an eccentric orbit is unable to gather a relevant amount of gas in its vicinity, having a much higher velocity relative to the background. Only when the MBH orbit becomes nearly circular, gas is collected close by. This may conduct to the formation of an accretion disc fueling the MBH, hence triggering nuclear activity, observable on scales of a few pc, during the pairing. Then, AGN activity could be linked to the dynamics of the pairing process of the MBHs inside circum– nuclear discs. The possible presence of accretion discs near the horizons of coalescing MBHs is of particular importance, as it may leave an electromagnetic signal correlated the GW emission targeted by LISA (Armitage & Natarajan 2002, Milosavljevíc & Phinney 2005, Kocsis et al. 2005). We plan to improve upon our model, e.g., including gas cooling and heating, star formation, BH treated as sink particles, in order to explore the accretion issue in much greater detail. Figure 14. Same as Figure 3 for run F. Solid (dashed) line refers Whether circularization stalls the LISA BHs to separa- to the lighter (heavier) BH. tions larger than critical for the intervention of GWs is not clear yet. The exploration of later stages requires a much higher force resolution and a modeling of the BHs as “ab- sorbing” particles, having an horizon, i.e., a trapping sur- 7 binary with a low eccentricity. However, the numerical noise face. ELCM05 show, for two MBH = 5 × 10 M⊙, that the at the end of the simulation does not allow us to calculate dynamical action of the torque continues at least down to a the precise value of any residual eccentricity. In the case of separation ≃ 0.1 pc, where the time to coalescence (because counter–rotating orbits, the eccentricity does not decrease of gravitational wave emission) is about 10 Gyrs. However since the density wake remains always behind the BH mo- it is not clear if the process is still relevant for different tion, and the MBHs may end forming a binary with still conditions, not discussed by ELCM05, i.e., different disc– significant eccentricity (Figure 12). The probability of pair- to–binary mass ratio, and/or different BH–BH mass ratio. ing along counter or co–rotating orbits is not known yet, We can just note that binary BHs in gaseous rotating back- and should be further investigated creating a statistically ground do not bind on those highly eccentric plunging or- significant sample of simulations of gas–rich merging galax- bits that would bring them straight to coalescence. LISA ies, similar to those carried out by Kazantzidis et al. (2005). BHs form circular binaries and may need additional mech- To summarize, if co–rotating orbits are more likely, we anisms for continuing their hardening process. Three–body can note that the braking of MBHs in a gaseous back- encounters with low angular momentum bulge stars is a pos- ground deliver a MBHB on a nearly circular orbit. It is worth sibility (Mikkola & Valtonen 1992, Quinlan 1996), though it noticing that, during later phases, further eccentricity evolu- seems that BH binaries must be on highly eccentric orbits tion may still occur, driven by close encounters with single to drive the separation to sub-pc scales, where gravitational stars (Mikkola & Valtonen 1992, Quinlan 1996, Milosavl- wave emission takes over (Sesana et al. 2005, in preparation). jevic & Merritt 2001, Aarseth 2003, Berczik, Merritt & One aspect worth considering is that, during the hardening Spurzem 2005), and/or gas–dynamical processes (Armitage phase, one or both BHs could accrete gas, hence increase & Natarajan 2005), before GW emission acts to circularize their mass, modifying the dynamics of the binary. Lets us the orbits. In Armitage & Natarajan (2005) the eccetricity is now suppose that the mass of the member M1 (M2) of a shown to grow substantially at very small separations, when circular binary increases by a factor β1 (β2). Then, if the bi- gravitational torques from the MBHB act to clear a gap in nary angular momentum is conserved, we have a reduction the circum–binary disc. This fact may have important con- of the binary separation by a factor sequences for a possible final, GW driven, coalescence of the anew β1M1 + β2M2 two BHs. = 2 2 . (4) aold β β (M1 + M2) We find in addition, that when dynamical friction has 1 2 subsided, in the case of equal mass BHs, the binary that As an example, let us consider a very unequal mass binary forms is surrounded by gaseous particles belonging to the (M1 ≫ M2): the heavier BH can indeed accrete while the head of the wakes that merge in a coherent pattern, i.e., an secondary is slowly spiraling inwards, as in run D. From ellipsoidal mass distribution. In the case of unequal masses, equation 4, an increase of M1 by a factor of, say, 2, re- we observe on the contrary that the trailing over-density duces the binary separation by the same factor. This is quite created by the light BH brings it to a closer distance from the promising, as, in scattering experiments (Sesana et al. 2005, heavier central BH. Given the large unbalance between the in preparation), LISA circular binaries tend to stall at sep-

c 0000 RAS, MNRAS 000, 000–000 LISA double black holes 11 arations which are factors ∼ 2 − 8 (depending on mass and Sesana A., Haardt F., Madau P., 2005, in preparation mass ratio) too large to drive the binary to coalesce within Shakura N.I., Sunyaev R.A., 1973, A&A, 24, 337 1 Gyr because of gravitational wave emission. Soifer B.T. et al. , 1984, ApJ, 278, L71 Springel V., Yoshida N., White S.D.M., 2001, NewA, 6, 79 van den Bosch F.C., Lewis G.F., Lake G., Stadel J., 1999, ApJ, 515, 50 ACKNOWLEDGMENTS van der Marel R.P., Gerssen J., Guhathakurta P., Peterson R.C., Gebhardt K., 2002, AJ, 124, 3255 The authors thank Andres Escala, Lucio Mayer, Ruben Sal- Volonteri M., Haardt F., Madau P., 2003, ApJ, 582, 559 vaterra, Javier Sanchez, Boris Sbarufatti and Alberto Sesana Wada K., Norman C.A., 2001, ApJ, 546, 172 for fruitful comments and suggestions, and Franz Livio and Yu Q., 2002, MNRAS, 331, 931 Luca Paredi for technical support.

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